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Structure of the space of GL4 (Z2)-coinvariants Z2 ⊗ GL4 (Z2) P H∗ (Z42,Z2) in some generic degrees and its application
  • Đặng Võ Phúc
Đặng Võ Phúc
University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa, University of Khanh Hoa

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Let $A$ denote the Steenrod algebra at the prime 2 and let $k = \mathbb Z_2.$ An open problem of homotopy theory is to determine a minimal set of $A$-generators for the polynomial ring $P_q = k[x_1, \ldots, x_q] = H^{*}(k^{q}, k)$ on $q$ generators $x_1, \ldots, x_q$ with $|x_i|= 1.$ Equivalently, one can write down explicitly a basis for the graded vector space $\pmb{Q}^{q} := k\otimes_{A} P_q$ in each non-negative degree $n.$ This is the content of the classical “hit problem” in literature \cite{F.P}. Based on this problem, we are interested in the $q$-th algebraic transfer $Tr_q^{A}$ of W. Singer \cite{W.S1}, which is one of the useful tools for describing mod-2 cohomology of the algebra $A.$ This transfer is a morphism from the space of $G(q)$-coinvariant $k\otimes _{G(q)} P_A((P_q)_n^{*})$ of $\pmb{Q}^{q}$ to the $k$-cohomology group of the Steenrod algebra, ${\rm Ext}_{A}^{q, q+n}(k, k).$ Here $G(q)$ is the general linear group of degree $q$ over the field $k,$ and $P_A((P_q)_n^{*})$ is the primitive part of $(P_q)^{*}_n$ under the action of $A.$ Singer conjectured that $Tr_q^{A}$ is a monomorphism, but this remains unanswered for all $q\geq 4.$ The present paper is devoted to investigating Singer’s conjecture for rank 4. More specifically, basing the techniques of the hit problem of four variables, we explicitly determine the structure of $k\otimes _{G(4)} P_A((P_4)_{n}^{*})$ in some generic degrees $n.$ Applying these results and a representation of $Tr_4^{A}$ over the lambda algebra, we notice that Singer’s conjecture is true for rank $q = 4$ in respective degrees $n.$ This has contributed to the final proof of Singer’s conjecture in the rank 4 case.